Mechanical modeling and frequency splitting mechanism analysis of variable density hemispherical resonators

Mechanical processing errors can trigger the uneven mass and stiffness of the hemispherical resonator, leading to the generation of frequency splitting, which reduces the vibration performance of the hemispherical resonator and affects the output accuracy of the gyroscope. Therefore, analysing the influence mechanism of each harmonic of the mass distribution on the frequency splitting is crucial for gyro accuracy improvement. In this paper, the hemispherical resonator spherical shell is simplified into the form of an elastic ring, and based on Hamilton's principle and the Bubnov-Galliakin method, a ring dynamics model of the resonator in the form of a variable density is established, which is different from that of the resonator dynamics model that has already been established so far, which can only analyze the mechanism of the influence of the fourth harmonic of the mass defect on the frequency splitting. The model developed in this paper can analyze the influence mechanism of mass defects' first four harmonics on the frequency splitting, revealing the mathematical relationship between the frequency splitting and the amplitude of each harmonic component. It is theoretically demonstrated that the effect of the first three harmonics of a mass defect on frequency splitting is an order of magnitude smaller than the fourth harmonic of a mass defect. To verify the correctness of the theoretical model, an ion-beam etching system is built to complete the resonator circumferential harmonic mass removal experiments of various orders. The experiments show that when the etching time is 150 minutes, the resonator changes from an initial frequency splitting of 0.1158 Hz to 0.2503 Hz after the first harmonic etching, 0.3867 Hz after the second harmonic etching, 0.5221 Hz after the third harmonic etching, and 2.867 Hz after the fourth harmonic etching. It can be seen that the effect of the fourth harmonic of the mass defect on the frequency splitting is one order of magnitude higher than the first three harmonics, which is in general agreement with the results of the theoretical analysis.